Integral Operators With Rational Kernels

Abstract

ABSTRACT: Let q(s,t)=a+b(s+t)+cst+d(s²+12), where a, b, c, d e R, not all zero, 1 and suppose that k =-. A closed (bounded) interval I is called admissible for k if q(s,t) ≠0, for all s, te I, in which case we obtain a compact symmetric integral operator K, on L²(I) with continuous kernel k: K₁f(s)=(s K₁f(s) = f(t) dt q(s,t) (feL²(I), se I). In Chapter 2 we find the number of positive and the number of negative eigenvalues of K, in the case d = 0, and in Chapter 4 we do the same for a general d. In Chapter 3 we find the number of positive and the number of negative eigenvalues of integral operators of kernels of the form A+B(s+t)+Cst D+E(s+t)+Fst. In Chapter 5 we return to the case k = and d = 0 and we find asymptotic estimates 1 9 for the positive eigenvalues of K, and the negative eigenvalues